302 ELECTRICAL APPARATUS
or inductively, in series with each other, to an alternating vol-
tage—alternating-current motors with series-motor characteristic.
Herefrom then follow three main classes of alternating-current
motors:
Synchronous motors.
Induction motors.
Commutator motors.
There are, however, numerous intermediate forms, which
belong in several classes, as the synchronous-induction mo.tor,
the compensated-induction motor, etc.
172. An alternating current, I, in an electric circuit produces
a magnetic flux, i>, interlinked with this circuit. Considering
equivalent sine waves of I and <i>, $ lags behind I by the angle
of hysteretic lag, a. This magnetic flux, $, generates an e.m.f.,
E = 2irfn3>, where / = frequency, n = number of turns of
electric circuit. This generated e.m.f., E, lags 90° behind the
magnetic flux, $, hence consumes an e.m.f. 90° ahead of $,
or 90—a degrees ahead of I. This may be resolved in a reactive
component: E''=• 2-jrfn$ cos a = 2 irfLI = xl, the e.m.f. con-
sumed by self-induction, and power component: E" = 2irfn$
sin a = 2 irfHI = r"I = e.m.f. consumed by hysteresis (eddy
currents, etc.), and is, therefore, in vector representation denoted
by:
Er = jxl and E" = r"/,
where:
x = 2 TT/L = reactance,
and
L = inductance,
r" = effective hysteretic resistance.
The ohmic resistance of the circuit, /, consumes an e.m.f.
r'f, in phase with the current, and the total or effective resistance
of the circuit is, therefore, r = r' + r", and the total e.m.f.
consumed by the circuit, or the impressed e.m.f., is:
# = (r+ **)/ = 27,
where:
Z — r + jx = impedance, in vector denotation,
z == Vr2 + z2 = impedance, in absolute terms.
If an electric circuit is in inductive relation to another electric
circuit, it is advisable to separate the inductance, L, of the cir-